On a family of singular continuous measures related to the doubling map

TL;DR

This paper investigates a family of singular continuous measures linked to the doubling map, demonstrating their extremal properties and super-polynomial asymptotics, including the Thue--Morse measure.

Contribution

It introduces a new family of singular continuous measures represented by Riesz products, highlighting their extremal behavior and asymptotic properties.

Findings

Measures are purely singular continuous

Distribution functions exhibit super-polynomial asymptotics

Includes the Thue--Morse measure as a special case

Abstract

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue--Morse measure.